A study of wave propagation in varying cross-section waveguides by modal decomposition. Part I. Theory and validation

Abstract

The propagation of acoustic waves in waveguides with variable cross section is considered using multimodal decomposition. The approach adopted is to construct two infinite first-order differential equations for the components of the pressure and the velocity projected over the normal modes. From these an infinite matricial Riccati equation is derived for the impedance matrix. These equations are ordinary differential equations that can be integrated after truncation at a sufficient number of modes and take into account the coupling between modes. The stiffness of the pressure-velocity equations induced by the presence of evanescent modes is avoided by first calculating the impedance matrix along the guide. The method is checked using different examples where the solutions of the plane-wave approximation or the finite element method are known. Results show the method provides simple and accurate means to obtain the acoustic field with correct boundary conditions in a nonuniform guide with no restriction on the flare.

Broadband Low-Frequency Electroacoustic Absorbers Through Hybrid Sensor-/Shunt-Based Impedance Control

“…As shown in [16], in the case of a hard-wall termination, the resonance frequencies are equal to f rn = n c 2L (15) where n ∈ N. The anti-resonance frequencies f arn can be computed from (14) when…”

“…If the absorber area was substantially smaller than the cross-section of the waveguide, the hypothesis of a uniform pressure at the boundary would not be valid anymore. The analytical approach would require the total decomposed field on transverse modes [15], whereas a practical approach could be done with the help of a Finite Element Method software [16]. A surface impedance condition is imposed at the right end of the waveguide.…”

Improved multimodal method for the acoustic propagation in waveguides with a wall impedance and a uniform flow

“…Guided waves have been subject to investigation for many years and they have a wide range of applications in engineering and physics [1,2] in the context of electromagnetic, [3,4] in acoustics, [5,6] in elasticity and [7][8][9] in the context of shallow oceans. For uniform and infinite waveguides, separation of variables is possible, and exact solutions of the guided wave propagation (called modes) exist.…”

Effect of Cross-Sectional Area Discontinuities in Closed Hard-Walled Ducts on Frequency of Longitudinal Modes

“…In some systems, variations in duct size are accompanied by rapid changes in duct shape causing a formation of resonant structure resembling closed duct with cross-sectional discontinuities. A behaviour of sound waves in duct discontinuities has been studied with the aid of numerous methods (Miles, 1946a(Miles, , 1946bKaral, 1953;Kergomard, Garcia, 1987;Sahasrabudhe, Munjal, 1995;Pagneux et al, 1996;Muehleisen, Swanson, 2002). For low frequencies an analysis, using only planar zero order mode, is employed.…”